If each of $A,$ $B,$ $C$ stands for a nonzero digit in the following addition, then find the three-digit number $ABC.$

\[
\begin{array}{cccc}
& A & C & B \\
& B & A & C \\
& B & C & A \\
& C & A & B \\
+ & C & B & A \\ \hline
1 & 2 & 2 & 3
\end{array}
\]